Any planar graph on \(n\)  vertices that each have degree at most 4 can be drawn on an \(n \times n\)  grid such that the edges are non-intersecting grid paths [1]}.
It follows that there exists such an embedding on a grid with side length \(4n\)  for \(G^{\prime }\) .
Notice that \(G^{\prime }\)  is also a minor of this grid, so we conclude that also \(G\)  is a minor.
